ON THE BOUNDEDNESS OF MOTIONS OF MECHANICAL SYSTEMS DESCRIBED

BY FUZZY ORDINARY DIFFERENTIAL EQUATIONS

A. A. Martynyuk and V. I. Slyn’ko UDC 536.36

A new approach is proposed for analyzing the dynamics of fuzzy systems. Within the framework of the

approach, boundedness conditions for the solutions of fuzzy systems are established on the basis of the

classical inequalities of convex geometry

Keywords: fuzzy mechanical systems, boundedness, Brunn–Minkowsky inequality

Introduction. In the last three decades, interest to fuzzy dynamic systems has steadily heightened. The paper [12]

initiated the development of a set theory currently known as the theory of fuzzy sets. Later this theory was applied to some

problems in modern natural sciences, in particular, in control theory. Despite the success in the development of the theory of

fuzzy dynamic systems [5, 6, 12, 13], many problems of the qualitative theory of fuzzy differential equations remain unresolved.

One of such problems is to establish criteria for the boundedness of the motions of mechanical systems described by fuzzy

ordinary differential equations. Some results on the subject are available in [11].

To solve this problem, we develop a new approach based on the decomposition, proposed in [8], of a space E n into two

spaces: a Euclidean space R n and a quotient space E E Rn n nρ = / , followed by transformation of the original fuzzy ordinary

differential equation. The boundedness criteria are based on the existence of an auxiliary function and a functional both

satisfying special conditions and are illustrated by an example of a system defined in the space E 2 .

1. Supplementary Results. Let E n be a space of fuzzy sets obeying the following axioms:

F1. Sets [ ] | ( ) u x R u xnα α= ∈ ≥ are nonempty, closed, and bounded in R n for allα ∈ (0, 1];

F2 . A set [ ] [ ]u cl u00= ∪ >α

α is bounded;

F3 . u is fuzzy and convex, i.e., [ ]u x y u x u y( ( ) ) min ( ), ( )λ λ+ − ≥1 for all λ ∈ [0, 1], x y R n, ∈ .

In the space KCn of convex closed compacts R n , we introduce a Hausdorff metric [4] d A BH ( , )and Hausdorff distance

δH A B( , ) as

d A B R x B R y AHx A y B

( , ) max sup ( , ), sup ( , )=⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪∈ ∈,

where R x A x yy A

( , ) | | | |= −∈inf , | |. | | is the Euclidean norm in R n ,

δHx R

HA B d A x Bn

( , ) ( , )= +∈inf .

In the space E n , we introduce a Hausdorff metric DH ( , )⋅ ⋅ as D u v u vHl

( , ) ([ ] , [ ] )=∈

supα

α α . The operations of summation

and multiplication by a non-negative scalar are defined in the space E n [7]. It is well known that ( , )E DnH is a complete metric

International Applied Mechanics, Vol. 41, No. 12, 2005

1063-7095/05/4112-1407 ©2005 Springer Science+Business Media, Inc. 1407

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya

Mekhanika, Vol. 41, No. 12, pp. 93–99, December 2005. Original article submitted July 26, 2004.

space. Let E E Rn n nρ = / be a quotient space,Φ:E En n→ ρ be a canonical homomorphism, andΨ:E En n

ρ → be the mapping of

choice. In the space E nρ , we introduce the operations of summation and multiplication by a non-negative scalar and a metric

d ( , )⋅ ⋅ as follows:

U V U V+ = +Φ Ψ Ψ( ( ) ( )), U V E n, ∈ ρ ,

λ λU U=Φ Ψ( ( )), λ ≥ 0, U E n∈ ρ ,

d U V D U V xx R

Hn

( , ) inf ( ( ), ( ) )= +∈

Ψ Ψ ,

where ( , )E dnρ is a complete metric space.

Introduce a mappingΠ( ) ( , )un

h p pd dn

u

S n

=−∫∫ω

α α ω10

1

(whereω n is the area of a unit sphere S dn−1 , ω is an area element

of this sphere, and h pu ( , )α is the support function of a fuzzy set u E n∈ ) and a mappingΧ →:E En nρ ,Χ = −U U UΨ Π Ψ( ) ( ( )). In

the space E En n× ρ , we introduce the operations of summation and multiplication by a non-negative scalar and a metric:

( , ) ( , ) ( , ), ( , ) ( , )x U y V x y U V x U x U+ = + + =λ λ λ , D x U y V x y d U V(( , ), ( , )) | | | | ( , )= − + .

The mapping Β × →:R E En n nρ , Β ΧΨ( , ) ( )x U x U= + implements an isomorphism of the spaces R En n× ρ and E n , the

metrics DH and D being equivalent.

In the space E n , we consider the following fuzzy ordinary differential equation (ODE):

du

dxf t u u t u= =( , ), ( )0 0 , (1.1)

where u E n∈ , f Lip T E En n∈ ×( , )0 , and t T R∈ ⊂ +0 . The following system of equations is equivalent to Eq. (1.1) in the

product space R Enpn× :

dx

dtF t x U x t x

dU

dtG t x U U t U= = = =( , , ), ( ) , ( , , ), ( )0 0 0 0 , (1.2)

where ( , )x U R En n∈ × ρ , t T R∈ ⊂ +0 , F T R E Rn n n: 0 × × → , G T R E En n n: 0 × × →ρ , and F t x U f t x U( , , ) ( , ( , ))= Β−Π 1 ,

G t x U f t x U( , , ) ( , ( , ))= Β−Φ 1 .

Definition 1.1.

(i) The motion described by the function u t t u( ; , )0 0 in the fuzzy ODE (1.1) is called bounded if and only if there exists a

constant K K t u= ( , )0 0 such that D u t t u KH ( ( ; . ), )0 0 0 ≤ for all t t t J t u∈ +[ , ( , ))0 0 0 0 .

(ii) The motion x t t x U U t t x U( ; , , ), ( ; , , )0 0 0 0 0 0 of the dynamic system (1.2) is called x-bounded (U-bounded) if and

only if there exists a constant K K t x U= >* ( , , )0 0 0 0 such that | | ( ; , , )| | *x t t x U K0 0 0 ≤ (d U t t x U K( ( ; , , ), )0 0 0 0 ≤ ), where

[ , ( , ))t t J t u0 0 0 0+ is the maximum interval in which the solution of the fuzzy ODE (1.1) exists.

Note that due to the equivalence of the metrics DH and D, the x-boundedness and U-boundedness of the solution

( ( ; , , ), ( ; , , )x t t x U U t t x U0 0 0 0 0 0 ) of system (1.2) are obviously equivalent to the boundedness of the solution u t t u( ; , )0 0 ,

u x U0 0 0=Β( , ) of the fuzzy ODE (1.1).

Let us introduce similar definitions for the comparison system

dw

dtg t w w w t w

11 1 2 1 0 10= =( , , ), ( ) ,

dw

dtg t w w w t w

22 1 2 2 0 20= =( , , ), ( ) (1.3)

1408

whose right-hand sides are continuous and satisfy the Chaplygin–Wazewski conditions with respect to the variables w1 and w2.

Let w t t w+ ( ; , )0 0 be the upper solution of system (1.3).

Definition 1.2.

(i) The solution w t t w+ ( ; , )0 0 , w0 >0, of the comparison equations (1.3) is called bounded if and only if there exists a

constant K K t w= 1 0 0( , ) such that w t t w K e+ <( ; , )0 0 1 , e= Τ( , )1 1 for all t t≥ 0 .

(ii) The solution (w t t w w w t t w w1 0 10 20 2 0 10 20+ +( ; , , ), ( ; , , )) of system (1.3) is called w1-bounded (w2-bounded) if and

only if there exists a constant K K t w= >( , )0 0 0 such that 0 1 0 10 20 1≤ ≤+w t t w w K( ; , , ) * (0 2 0 10 20 1≤ ≤+w t t w w K( ; , , ) * ) for all

t t≥ 0 .

2. Basic Results. Let us formulate the boundedness conditions for the solutions of the fuzzy ODE (1.1) based on the

decomposed system (1.2).

Theorem 2.1. Suppose that the system of equations (1.2) is such that:

(i) there exist a function W T R Rn1 0: × → + and a functional W T E Rn

2 0: × → +ρ that are locally Lipschitz in x and U,

respectively;

(ii) there exist comparison functions a1 (. ) and a2 (. ) such that a ri ( )→∞ as r→∞ and

a x W t x1 1(| | | | ) ( , )≤ for all ( , )t x T R n∈ ×0 ,

a d U W t U2 20( ( , )) ( , )≤ for all ( , )t U T E n∈ ×0 ρ ;

(iii) there exist functions g C T R R R ii ∈ × × =+ + +( , ), ,0 1 2, such that

D W t x g t W t x W t x+ ≤1 1 1 2( , ) ( , ( , ), ( , )),

D W t x g t W t x W t x+ ≤2 2 1 2( , ) ( , ( , ), ( , ));

(iv) the system of equations

dw

dtg t w w w t w

dw

dtg t w w w

11 1 2 1 0 10

22 1 2 1= = =( , , ), ( ) , ( , , ), (t w0 20)=

satisfies the Chaplygin–Wazewski conditions with respect to the variables ( , )w w1 2 for all t T∈ 0 ;

(v) the upper solution of the comparison system (1.3) is w1-bounded (w2 -bounded). Then the solution

( ( ; , , ), ( ; , , )x t t x U U t t x U0 0 0 0 0 0 ) of system (1.2) is x-bounded (U-bounded).

Proof. By virtue of the comparison principle, we have the estimates

W t x t t x U w t t W t x W t x1 0 0 0 1 0 1 0 0 2 0 0( ; ( ; , , )) ( ; , ( , ), ( , ))≤ + ,

W t x t t x U w t t W t x W t x1 0 0 0 1 0 1 0 0 2 0 0( ; ( ; , , )) ( ; , ( , ), ( , ))≤ + .

Suppose that the solution (x t t x U U t t x U( ; , , ), ( ; , , )0 0 0 0 0 0 ) is not x-bounded, i.e., there exists a sequence of moments

τ k t t J t x U∈ +[ , ( , , ))0 0 0 0 0 such that | | ( ; , , )| |x t x Ukτ 0 0 0 →∞ as k →∞. By virtue of condition (2), we have

W x t x Uk k1 0 0 0( , ( ; , , ))τ τ →∞, which contradicts the w1-boundedness of the solution of the comparison system.

Let us establish geometrical conditions for the boundedness of the solutions of fuzzy dynamic systems. We will now

look into the issue of U-boundedness.

Let us formulate the basic concepts of convex geometry, according to [1, 4]. Let A and B be closed convex bodies in the

space R n ; then the Steiner formula holds for the volumeV A B( )+δ [3, 4]:

V A B V A n V A B C V A B V Bnn( ) ( ) ( , ) ( , ) ( )+ = + + + +δ δ δ δ1

2 22 ,

where δ ≥ 0, V A B1 ( , ) is the first mixed volume, V A B2 ( , ) is the second mixed volume, etc. Here is the generalized

Steiner–Minkowski formula for the first mixed volume [3]:

1409

V A B K V A K n W C W V Bnn

1 1 11 12 2

111

11( , ) ( , ) ( ) (+ = + − + + +−−δ δ δ δ , )K ,

where δ ≥ 0 and W i ni1 12 2, , , ,= − , are mixed Minkowski integrals of cross-section measures, which have a well-known

geometrical meaning [4]. For example, nV A1 ( )is the surface area of the body A. We will further consider fuzzy sets u E n∈ such

that int[ ]u 0 ≠Θ if only [ ]u ≠ 0.

Theorem 2.2. Let the system of equations (1.2) be such that the following inequality holds:

V u K W u f t u V u V K Vn n1

1 011

0 0 2 01

2− −≤([ ] , ) ([ ] ,[ ( , )] ) ([ ] ) ( ) 1

0 0([ ] ,[ ( , )] )u f t u .

Then there exists a constant r0 >0 such that the solution ( ( ; , , ),x t t x U0 0 0 U t t x U( ; , , ))0 0 0 of system (1.2) is U-bounded

if only d U r( , )0 00 ≤ .

Proof. Consider a functional W E Rpn: → defined by the formula

W U V u K V u V K u Un n( ) ([ ] , ) ([ ] ) ( ), ( )= − =−1

0 1 01

2Ψ .

Obviously, the functional W is correctly defined since it is independent of the choice of the mapping Ψ and is

nonnegative by virtue of the classical Brunn–Minkowski theorem [6, 7].

It follows from Eq. (1.1) that u t h u t f t u h o h( ) ( ) [ ( , )] ( )+ = + +0 for all rather small h ≥ 0. Hence, by virtue of the Steiner

formulas, we get

V u t h V u t nV u t f t u h o([ ( )] ) ([ ( )] ) ([ ( )] , [ ( , )] ) (+ = + +0 01

0 0 h ),

V u t h K V u t K n W u t f10

10

1101([ ( )] , ) ([ ( )] , ) ( ) ([ ( )] , [ (+ = + − t u h o h, )] ) ( )0 + .

Formulas (2.1) allow us to evaluate the derivative of the functional using the system of equations (1.2):

dW

dtn n V u t K W u t f t un= − −( )( ([ ( )] , ) ([ ( )] ,[ ( , )] )1 1

1 011

0 0 − −1

22 0

10 0V u t V K V u t f t un ([ ( )] ) ( ) ([ ( )] ,[ ( , )] ).

It follows from Theorem 2.2 that W U t t x U W U( ( ; , , )) ( )0 0 0 0≤ for all t t t J t x∈ +[ , ( , ))0 0 0 0 . Therefore, the following

estimate holds:

V u V u V K W Un n1

0 1 00([ ] ) ([ ] ) ( ) ( ).− <− (2.2)

Applying the triangle inequality, we obtain

δ δ δH H Hu t u t sK sK([ ( )] , ) ([ ( )] , ) ( , )0 00 0≤ + ,

where sV u t

V K

n

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

([ ( )]

( )

/0 1

.

The results from [3, Theorem 2] allow us to specify an estimate δH u t sK([ ( )] , )0 ≤C W U n( ( )) /0

1 2(C is a positive

constant) that is valid for all t t t J t u∈ +[ , ( , ))0 0 0 0 if only W U( ) ,0 0 0<ε ε is a rather small positive constant.

Due to the well-known theorems of convex geometry [1, 4], the functional W U( ) is continuous; therefore, there exists a

positive constant r0 such that d U r( , )0 00 < if only W U( )0 0<ε . It is easy to verify that

δH

n n

nsK s

W U

V K( , )

[ ( )]

( )

/ ( )

/ ( )0

2 01 1

1 1= ≤

−

− .

Hence

1410

δH

n n

nu t

W U

V KC W U([ ( )] , )

[ ( )]

( )( ( ))

/ ( )

/ ( )0 0

1 1

1 1 002

< +−

−1 2/n .

By virtue of the well-known property of the Hausdorff metric d u x d u xH ([ ] , ) ([ ] , )α + ≤ +0 00 for all x R n∈ andα ∈ I,

we have d U uH( , ) ([ ] , )0 00≤δ . From here we derive the estimate

d U tW U

V KC W U

n n

nn( ( ), )

[ ( )]

( )( ( ))

/ ( )

/ ( )/0

2 01 1

1 1 01 2≤ +

−

−

for all t t t J t u∈ +[ , ( , ))0 0 0 0 , which proves Theorem 2.2.

Corollary. Let the system of equations (1.2) in the space R E2 2× be such that the inequality

l u l f t u S u f t u([ ] ) ([ ( , )] ) ([ ] ,[ ( , )] )0 0 0 02≤ π

holds for all u E∈ 2 , where l u([ ] )0 is the length of the zero-level boundary [ ]u 0 , and S ( , )⋅ ⋅ is the mixed Minkowski area [4].

Then the motions of system (1.2) are U-bounded if onlyU E p02∈ ⊂Ω is a rather small neighborhood of the zero of the

space E p2 .

Let us now establish boundedness conditions for the motions described by the system of equations (1.2).

Theorem 2.3. Let the system of equations (1.2) be such that the following conditions are satisfied:

(i) the inequality

V u K W u f t u V u V K Vn n1

1 011

0 0 2 01

2− −≤([ ] , ) ([ ] ,[ ( , )] ) ([ ] ) ( ) 1

0 0([ ] ,[ ( , )] )u f t u

holds for all u E un∈ ≠∅, [ ]int 0 ;

(ii) there exist a function W t x( , ) locally Lipschitz in x for all t T∈ 0 and a comparison function a( )⋅ such that a r( )→∞as r→∞ and a x W t x(| | | | ) ( , )≤ ;

(iii) the estimate D W t x g t W t x d U+ ≤( , ) ( , ( , ), ( , ))0 holds for all ( , , )t x U T R Enpn∈ × ×0 , where g t( , , )⋅ ⋅ is a function

monotonically nondecreasing for all t T∈ 0 ;

(iv) the upper solutions w t t w( ; , , )0 0 η of the family of comparison equations

dw

dtg t w w t w= =( , , ), ( )η 0 0 (2.3)

are bounded for rather small η> 0.

Then the motions described by the system of equations (1.2) are bounded for all η, η is a rather small neighborhood of

the zero of the space E pn .

Proof. According to Theorem 2.2, the solutions of system (1.2) are U-bounded if only U E pn

0 ∈ ⊂Ω ; therefore, the

inequality D W t x g t W t x+ ≤( , ) ( , ( , ), )η holds for rather small η> 0. Applying the comparison principle, we arrive at the estimate

W t x( , )≤ +w t t W t x( ; , ( , ), )0 0 0 η valid for all t t t J t x U∈ +[ , ( , , ))0 0 0 0 0 . Suppose that the motions described by the system of

equations (1.2) are not x-bounded, i.e., there exists a sequence ( , , )τ k k t J t x U=∞ → +1 0 0 0 0 such that | | ( ; , , )| |x t x Ukτ 0 0 0 →∞

as k →∞. Under the hypothesis of the theorem, W x t x Uk k( , ( ; , , ))τ τ 0 0 0 →∞as k →∞. This contradicts the boundedness of the

upper solution of the comparison equation (2.3), which proves the theorem.

3. An Example and Discussion of the Results. Let us consider the following fuzzy dynamic system in the space E 2 :

du

dt

t u

S uu t u u

m= = ≠∅

ψ( )

([ ] ), ( ) , [ ]

0 0 00int , (3.1)

where u E∈ 2 ; S u([ ] )0 is the zero-level area of the set u E∈ 2 ; andψ τ( )≥ 0, τ ≥ t0 .

1411

Applying the Steiner formula to Eq. (3.1), we get

S u t h S u t hS u tt u t

S um([ ( )] ) ([ ( )] ) [ ( ] ,

( )[ ( )]

([+ = +0 0 0

0

2ψ

( )] )( )

to h

0

⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

for rather small h; therefore,dS

dt

t

S m= −

2

1

ψ ( ).

Integrating this equation yields

S u t S u m dm m

t

t

([ ( )] ) ([ ] ) ( )00

0 2

0

= + ∫ ψ τ τ. (3.2)

Substituting (3.2) into (3.1), we find the solution of this equation in the form

u t S u m d um

t

t m

( ) ([ ] ) ( )

/

= +⎛

⎝

⎜⎜

⎞

⎠

⎟⎟∫0

0

1 2

02

0

ψ τ τ .

It is obvious that the solution u t( ) is bounded if ψ τ τ( )d

t

t

0

∫ <∞.

The approach proposed here to analyze the dynamics of fuzzy dynamic systems allows us to establish less conservative

conditions of boundedness of their solutions and to simplify the procedure of evaluating the total derivative of the auxiliary

function (an analog of the Lyapunov function) along the solutions of the system under consideration. Of interest is to discuss [9,

10] the possible application of matrix Lyapunov functions in the problem involved.

REFERENCES

1. W. Blaschke, Kreis und Kugel, Walter de Gruyter & Co, Berlin (1956).

2. V. I. Diskant, “Strengthening of an isoperimetric inequality,” Sib. Math. J., 14, No. 4, 608–611 (1973).

3. V. I. Diskant, “Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference,” Sib. Math.

J., 13, No. 4, 529–532 (1972).

4. H. Hadwiger, Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, Springer Verlag, Berlin (1957).

5. A. Kandel, Fuzzy Mathematical Techniques with Applications, Addison Wesley, New York (1986).

6. R. Kruse, J. Gebhardt, and F. Klawonn, Fuzzy-Systeme, Leitfaeden der Informatik, Stuttgart (1995).

7. V. Laksnmikantham and R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Florida Institute of

Technology, Melbourne (2003).

8. A. A. Ìàrtynyuk and V. I. Slyn’ko, “On a new approach in the theory of fuzzy differential equations,” Dokl. RAN, 402,

No. 3, 365–370 (2005).

9. A. A. Martynuyk and V. I. Slyn’ko, “Stability of a nonlinear impulsive system,” Int. Appl. Mech., 40, No. 2, 231–239 (2004).

10. A. A. Martynyuk and L. N. Chernetskaya, “A note on the theory of stability of impulsive systems with structural

perturbations,” Int. Appl. Mech., 39, No. 3, 350–355 (2003).

11. R. Mohapatra and Y. Zhang, “Criteria for boundedness of fuzzy differential equations,” Math. Inequal. Appl., 3, No. 3,

399–410 (2000).

12. M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York (1993).

13. L. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. SMC, 3,

28–44 (1973).

14. L. Zadeh, “Fuzzy sets,” Inf. Contr., No. 8, 338–353 (1965).

15. V. M. Shul’ga and O. M. Shul’ga, “Wave propagation in an orthotropic cylinder with a liquid,” Int. Appl. Mech., 39,

No. 6, 736–740 (2003).

1412